"Pope/bishop Telesphorus" is a fake, he really matches emperor Hadrian.
Some extra details matches evidences from their biographies:
Telesphorus is pretty certainly either/both Trajan &/or Hadrian. (Trajan is definitely "pope Alexander", and he may possibly also be either/both "pope Evaristus" &/or "pope Sixtus" &/or "pope Telesphorus". Though it looks like he was only Alex and not the others.) (Hadrian is pretty certainly any/all of either "pope Sixtus", "pope Telesphorus" &/or "pope Hyginus".)
-- Telesphorus: Name Telesphorus (tele "far", "accomplisher, bringer of completion", [*ter/trans/through "beyond"?]). -- Hadrian: i. "plans to complete temple of Zeus in Cyzicus"? "... he was able to see for himself the completed work of rebuilding the Pantheon. Also completed by then was Hadrian's villa...." "the almost complete [Hadrian's] mausoleum. Upon the completion of the tomb of Hadrian...." Hadrian never saw the finished Hadrian's wall? ii. [Hadrian spent half his reign traveling, and traveled to nearly every province.] iii. [wall controled cross-border trade & immigration. crossing Mediterranean to Mauretania. conveyed across Aegean.] iv. [Delphi??]
-- Telesphorus: "of Greek ancestry". -- Hadrian: Nicknamed 'Graeculus' "Greekling". Was a Phil-Hellene/Helleno-phile, "Hadrian's defined Greekness", ardent admirer of Greece, wanted to make Athens capital. He visited/travelled to Greece/Athens. Legio V Macedonica. The 'Adriatic Sea'?
-- Telesphorus: Was a glorious/great martyr. -- Hadrian: by inversion: Persecution of christians of Trajan & Hadrian (3rd of 10 major persecutions from Nero to Diocletian) [Justin Martyr??] '10 martyrs' (Judaic scholars). Bloody repression of Jewish revolt 132-6 (Roman looses heavy, an entire legion destroyed; 580000 Jews killed.) [1 whole legion disappeared/lost/destroyed in Britian in 119-121 during his reign?] [How many people died building Hadrian's Wall?]
-- Telesphorus: 7th/8th pope. -- Hadrian: 14th emperor. [Minus the 1st 4 emps, since Peter matches 5th emp Nero; & minus 3 "usurpers" Galba, Otho, Vitellius.]
-- Telesphorus: T.d.S., Calabria. -- Hadrian: Detour to Sicily?